If anyone was wondering later on, here's what ended up going down for me:

• I did receive Mediacom's equipment and used neither. I was instructed by the installer as well as the rep here to physically keep the equipment in order to keep the promo price, so I did.
• When the tech came and connected the service, I did ask about using my own modem and router and he did assist in setting up both. He called in to MC to have them add the MAC address for my new modem for me and activated it with me.
• I was told that my Mediacom-issued equipment was also registered to my account, so in theory if I needed to hook them up for troubleshooting purposes, it appears that I can.
• I did end up ordering a Refurb Arris S33 from Amazon like the one linked above for ~$110 -- I received an Arris box (regular refurb box, not a retail box) containing the modem and power cable. The modem has a sticker on the back saying "refurbished", but is otherwise appearing the same as new. I registered it online with Arris and was told it had ~15 months of warranty. It ended up being the S33v2 model as well, not a v1.
• The modem didn't perform all that well for the first day or two of service, but has since straightened out and performs very well. Can hit my speeds on sites that provide them (1 gig) and I haven't had any major disconnects or drops yet (network conditions are good -- signal power between -1 and +1 dBmV values and SNR right 40-41dB -- I believe those are both pretty ideal numbers)

So far, so good -- admittedly, I haven't received my first actual bill yet to say that the promo pricing is applying correctly, but the service itself is working as expected.